How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's say we work over an algebraically closed field of characteristic $p$. It seems to me that (motivated by the étale topology, say), authors tend to use the geometry of $\mathbb{C}$ as a template, envisioning $\mathbb{A}^1$ as something like the complex plane. Then, using this as a model, one accounts for extra phenomena that occur in characteristic $p$; namely, extra automorphisms due to frobenii, as well as some extra singularities that don't occur in characteristic $0$.

Is this picture of characteristic $p$ geometry accurate? To make this question a little less soft, I'll ask the following naive, and certainly false question: Is the category of varieties over $\overline{\mathbb{F}}_p$ equivalent to the category of varieties over $\mathbb{C}$ along with extra morphisms coming from Frobenii (as well as objects obtained from glueing via these extra morphisms)?